A weak measurement trades information per trial for reduced disturbance per trial. When the data are further conditioned on a chosen final state, the average pointer shift can be governed by a weak value that may lie far outside the ordinary eigenvalue range of the measured observable.
Guide 7 owns POVMs and generalised measurement. Guide 54 owns continuous filtering. This guide owns the preselected–postselected weak-measurement limit: von Neumann pointer coupling, weak values, anomalous shifts, success-probability accounting, signal amplification, weak-value tomography and the distinction between a conditional mean and a new eigenvalue.
Preselect |i⟩ → couple weakly to a pointer → postselect |f⟩ → condition the pointer statistics → weak value controls the leading shift.
1. Standard von Neumann measurement model
Let A be a system observable and let q,p be conjugate pointer variables with [q,p]=iℏ.
An impulsive measurement interaction can be written
U=exp(−ig A⊗p/ℏ).
If the system is in eigenstate |a⟩ with eigenvalue a, the pointer wavefunction shifts in q by ga.
A strong measurement uses sufficiently separated pointer distributions to resolve distinct eigenvalues in one or a few trials.
2. Weak coupling
In a weak measurement, g is small relative to the pointer width scale. The pointer distributions associated with different eigenvalues strongly overlap.
One trial reveals little information about A. Many repeated trials are needed to estimate the small mean pointer displacement.
The benefit is that the system-pointer entanglement and resulting disturbance can be small to leading order.
3. Preselection and postselection
Prepare the system in |i⟩, interact weakly with the pointer, then perform a final projective measurement and retain only trials with outcome |f⟩.
The unnormalised postselected pointer state is
|Φ_f⟩=⟨f|exp(−igA⊗p/ℏ)|i⟩|Φ⟩.
For sufficiently weak coupling, expand to first order in g.
4. Weak value
Using U≈I−igA⊗p/ℏ,
|Φ_f⟩≈⟨f|i⟩[I−ig A_w p/ℏ]|Φ⟩,
where
A_w=⟨f|A|i⟩/⟨f|i⟩.
A_w is generally complex.
It is undefined when ⟨f|i⟩=0 exactly; in that case the first-order expansion must be handled more carefully because the nominal postselection probability vanishes.
5. Real part shifts pointer position
For a centred real Gaussian pointer with ⟨q⟩=⟨p⟩=0 and no initial q–p correlation, the leading postselected mean position shift is
Δ⟨q⟩≈g Re(A_w).
Thus an anomalously large real weak value can produce a pointer displacement much larger than g times any eigenvalue of A.
6. Imaginary part shifts conjugate momentum
For the same Gaussian pointer, the imaginary part modifies the momentum distribution. To first order,
Δ⟨p⟩≈(2g/ℏ) Var(p) Im(A_w).
The exact coefficient depends on pointer convention and initial correlations, but the general result is that Re(A_w) and Im(A_w) govern different pointer responses.
7. Worked qubit weak value
Let A=Z. Preselect
|i⟩=|+x⟩=(|0⟩+|1⟩)/√2.
Postselect
|f⟩=cos(θ/2)|0⟩−sin(θ/2)|1⟩.
Then
⟨f|i⟩=[cos(θ/2)−sin(θ/2)]/√2
and
⟨f|Z|i⟩=[cos(θ/2)+sin(θ/2)]/√2.
Therefore
Z_w=[cos(θ/2)+sin(θ/2)]/[cos(θ/2)−sin(θ/2)].
As θ approaches π/2, the denominator becomes small and Z_w can become much larger than the eigenvalue range [−1,1].
8. Large weak value comes with rare postselection
Before the weak interaction, the nominal success probability is
P_f≈|⟨f|i⟩|².
The same nearly orthogonal choice that makes A_w large makes P_f small.
Thus a large shift among accepted trials is purchased by discarding many trials.
Signal amplification and information gain must therefore be evaluated with the postselection probability included.
9. Weak values are not new eigenvalues
If A has eigenvalues ±1 and A_w=20, this does not mean the observable acquired a physical eigenvalue 20.
A_w is a conditional amplitude ratio controlling a weak pointer response in a specified pre/postselected ensemble.
A subsequent strong measurement of A still yields only its ordinary eigenvalues.
10. Exact postselected pointer before approximation
If A has spectral decomposition A=Σ_a a|a⟩⟨a|, then
|Φ_f⟩=Σ_a ⟨f|a⟩⟨a|i⟩ exp(−iga p/ℏ)|Φ⟩.
This is a coherent superposition of pointer translations ga.
The weak-value approximation replaces this entire superposition by one effective small translation only when the translated pointer states overlap strongly enough.
11. Weak-regime validity parameter
A rough requirement is that the dimensionless coupling
g Δp |A_w|/ℏ
remain small enough that higher-order terms do not dominate the postselected pointer state.
Near nearly orthogonal postselection, |A_w| can be huge, so g may need to be correspondingly smaller. “Weak coupling” is therefore not determined by g alone.
12. Second-order corrections
The next term contains
(−g²p²/2ℏ²) (A²)_w
where (A²)_w=⟨f|A²|i⟩/⟨f|i⟩.
When the denominator is small, these higher weak moments can become important and deform the pointer distribution away from a simple translated Gaussian.
13. Strong-measurement limit
When g is large compared with the pointer width scale, translated pointer packets for different a become nearly orthogonal.
Interference between them becomes negligible, and the pointer behaves as a standard projective measurement of A.
Weak and strong measurements are therefore two regimes of one system–pointer interaction, not unrelated measurement postulates.
14. Weak measurement without postselection
If every final state is retained, the mean pointer shift is governed by the ordinary expectation ⟨A⟩ to leading order.
Weak values become distinctive when data are conditioned on a later outcome.
Postselection reorganises the ensemble, not the spectrum of A.
15. Signal amplification
If technical detector noise has a fixed scale independent of accepted sample count, a large weak-value pointer displacement can move a tiny signal away from an instrumental offset or digitisation floor.
This can be experimentally useful even when no fundamental Fisher-information gain occurs.
The correct comparison includes all attempted trials, accepted fraction, detector saturation, technical noise and the best non-postselected strategy allowed by the same hardware.
16. Fisher-information caveat
Postselection cannot generally create information from nothing.
When ideal shot noise and all trials are counted, the increased sensitivity per accepted event is often offset by the decreased success probability.
Weak-value amplification can still be advantageous under constrained detectors or particular technical-noise models, but those advantages are engineering statements with explicit assumptions.
17. Weak-value tomography
By weakly measuring projectors or other operators and postselecting in a complementary basis, one can reconstruct amplitudes or density-matrix elements from weak values.
Such protocols can make the reconstruction formula direct, but they still require repeated ensembles and calibration. They do not measure an unknown wavefunction from one individual system.
18. Direct wavefunction measurement claim
Experiments have used weak measurements and postselection to reconstruct a wavefunction without a traditional tomographic inversion step.
The word “direct” refers to the relationship between pointer averages and amplitudes, not to obtaining a complete wavefunction from one trial.
Statistical averaging remains essential.
19. Sequential weak values
Two or more weak interactions before postselection generate sequential or joint weak values involving ordered operator products.
Because noncommuting observables depend on order, the corresponding pointer correlations can encode temporal quantum structure.
This provides a bridge toward Guide 84’s temporal-correlation tests.
20. Weak measurement and contextuality
Anomalous weak values have been connected to contextuality under specified operational assumptions.
This does not mean every weak value outside an eigenvalue range is by itself a context-free proof of contextuality. The theorem depends on the measurement model, disturbance bounds and operational equivalences.
Guide 56 owns the broader contextuality framework.
21. Weak measurement and Bohmian-style trajectory reconstruction
Weak values of momentum conditioned on position can be used to reconstruct average flow lines associated with optical or quantum probability currents.
Calling these lines literal particle trajectories adds an interpretation beyond the measured conditional averages. The operational data are pointer shifts and postselection statistics.
22. Negative weak probabilities
Weak values of projectors can be negative or exceed one.
They are not ordinary Kolmogorov probabilities. They behave more like quasiprobabilities or conditional amplitudes and can encode interference between alternatives.
This connects naturally to Guide 79’s quasiprobability viewpoint.
23. Disturbance never vanishes exactly at finite information gain
A finite coupling entangles system and pointer and therefore causes some backaction in general.
The weak-measurement limit makes the disturbance small per trial while also making the information small per trial.
Calling a weak measurement “noninvasive” requires an operational tolerance and should not be interpreted as literally zero disturbance.
24. Common misconceptions
“A weak value is an eigenvalue.” It is a conditional amplitude ratio governing a pointer shift.
“A huge weak value gives huge free sensitivity.” Large values generally come with low postselection probability and a stricter weak-coupling requirement.
“Weak measurement causes zero disturbance.” The disturbance is small, not generically zero.
“Direct wavefunction measurement needs one particle.” The pointer statistics still require an ensemble.
25. Worked synthesis problem
Let A=Z, preselect |+x⟩ and choose θ so that tan(θ/2)=0.9 in the postselection of Section 7.
Divide numerator and denominator by cos(θ/2):
Z_w=(1+0.9)/(1−0.9)=19.
The nominal postselection amplitude is proportional to 1−0.9=0.1, so the success probability is small after proper normalisation.
A pointer can therefore shift by approximately 19g in the weak regime even though strong Z measurements return only ±1. The amplified conditional shift must be analysed together with the rare acceptance rate and higher-order corrections.
26. Practice set
- Write the von Neumann interaction used here.
- Define a weak value.
- What does Re(A_w) do to a centred Gaussian pointer?
- What does Im(A_w) do?
- Why can A_w lie outside the eigenvalue range?
- Why are large anomalous values often rare?
- What limits the first-order weak-value approximation?
- What happens in the strong-coupling limit?
- Why is weak-value amplification not automatically a Fisher-information advantage?
- Why are weak projector values not ordinary probabilities?
Answers
exp(−igA⊗p/ℏ).⟨f|A|i⟩/⟨f|i⟩.- It gives the leading q shift
gRe(A_w)under the stated pointer conditions. - It produces a conjugate-momentum shift proportional to the initial momentum variance.
- It is a ratio of transition amplitudes rather than a spectral eigenvalue.
- The denominator ⟨f|i⟩ becomes small, lowering postselection success probability.
- Higher powers of g p A and large weak moments must remain negligible.
- Pointer packets resolve different eigenvalues and the measurement becomes effectively projective.
- The accepted-sample shift can be offset by discarded trials under ideal shot-noise accounting.
- They can be negative or exceed one because they are quasiprobabilistic conditional quantities.
Sources and further study
[1] Yakir Aharonov, David Z. Albert and Lev Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Physical Review Letters 60, 1351 (1988). Foundational anomalous weak-value paper.
[2] A. G. Kofman, S. Ashhab and Franco Nori, Nonperturbative theory of weak pre- and post-selected measurements, Physics Reports 520, 43–133 (2012). Weak and beyond-weak regimes.
[3] Jeff S. Lundeen and colleagues, Direct measurement of the quantum wavefunction, Nature 474, 188–191 (2011). Weak-value-based reconstruction.
[4] Andrew N. Jordan, Julián Martínez-Rincón and John C. Howell, Technical advantages for weak-value amplification: when less is more, Physical Review X 4, 011031 (2014). Technical-noise/resource discussion.
[5] Matthew F. Pusey, Anomalous weak values are proofs of contextuality, Physical Review Letters 113, 200401 (2014), under its stated operational assumptions.
Continue through Quantum Mathematics — Batch 21
Guide 81: Quantum Darwinism, Pointer States, Environment as Witness, Redundancy and Decoherence develops environmental records. Guide 82: Quantum Discord, Classical–Quantum States, Conditional Entropy and Measurement Disturbance develops nonclassical mixed-state correlations. Guide 84: Leggett–Garg Inequalities, Macrorealism and Temporal Quantum Correlations develops temporal tests where measurement disturbance is a central assumption.